Periods of Real Biextensions

A real biextension is a real mixed Hodge structure that is an extension of R(0) by a mixed Hodge structure with weights $-1$ and $-2$. A unipotent real biextension over an algebraic manifold is a variation of mixed Hodge structure over it, each of whose fibers is a real biextension and whose weight graded quotients are do not vary. We show that if a unipotent real biextension has non abelian monodromy, then its ``general fiber'' does not split. This result is a tool for investigating the boundary behaviour of normal functions and is applied in arXiv:2408.07809 to study the boundary behaviour of the normal function of the Ceresa cycle.